8 CNN_Mathematical_Explanation.ipynb

Kernel: Python 3 (ipykernel)

Convolutional Neural Networks (CNN)

Step-by-Step Mathematical Explanation

This notebook explains CNNs with mathematical intuition and simple examples.

1. What is a Convolution?

A convolution is a mathematical operation between an input matrix (image) and a kernel (filter).

Formula:

2. Example Input Image and Kernel

We use a 3×3 image and a 2×2 kernel to explain convolution.

import numpy as np

image = np.array([[1, 2, 3],
                  [4, 5, 6],
                  [7, 8, 9]])

kernel = np.array([[1, 0],
                   [0, -1]])

image, kernel

(array([[1, 2, 3],
        [4, 5, 6],
        [7, 8, 9]]),
 array([[ 1,  0],
        [ 0, -1]]))

3. Convolution Calculation (Manually)

Each output value is computed by element-wise multiplication followed by summation.

output = np.zeros((2,2))
for i in range(2):
    for j in range(2):
        region = image[i:i+2, j:j+2]
        output[i,j] = np.sum(region * kernel)
output

array([[-4., -4.],
       [-4., -4.]])

[1,1]  f(x)= x*filter*w+b

4. Activation Function (ReLU)

ReLU introduces non-linearity.

relu_output = np.maximum(0, output)
relu_output

array([[0., 0.],
       [0., 0.]])

2*2
(0,0,0,0): 4 

5. Pooling (Max Pooling)

Pooling reduces spatial dimensions.

import numpy as np
b=[1,2,89]
c=np.array(b)
p=np.max(c)
p

89

pooled = np.max(relu_output)
pooled

6. Fully Connected Layer (Mathematics)

Flatten the feature map and apply:

Where:C = number of classes

x = relu_output.flatten()
W = np.array([0.5, -0.2, 0.1, 0.4])
b = 0.1
y = np.dot(W, x) + b
y

7. Summary

• Convolution extracts features

• ReLU adds non-linearity

• Pooling reduces size

• Fully connected layers perform classification

A CNN learns convolutional filters that extract hierarchical spatial features and optimizes them end-to-end using backpropagation to minimize classification loss.